Maximal antichains of subsets II: Constructions

2021/06/04 by Griggs, Jerrold R., Kalinowski, Thomas, Leck, Uwe +2
#05D05 (Secondary) #06A07 (Primary) #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.02230

Abstract

This is the second in a sequence of three papers investigating the question for which positive integers m there exists a maximal antichain of size m in the Boolean lattice B_n (the power set of [n]:=\1,2,…,n\, ordered by inclusion). In the previous paper we characterized those m between \binomn\lceil n/2\rceil-\lceil n/2\rceil² and the maximum size \binomn\lceil n/2 \rceil that are not sizes of maximal antichains. In this paper we show that all smaller m are sizes of maximal antichains.

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